**Description: **Axiom of Specialization.
A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all *x*, it is true for any
specific *x* (that would
typically occur as a free variable in the wff
substituted for *φ*).
(A free variable is one that does not occur in
the scope of a quantifier: *x* and *y* are both free in *x* = *y*,
but only *x* is free in
∀*y**x* = *y*.) Axiom scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a
weaker inference form of the converse holds and is expressed as rule
ax-gen 1253. Conditional forms of the converse are given
by ax-12 1318,
ax-15 1894, ax-16 1553, and ax-17 1331.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from *x* for the special case. For use, that
requires the assistance of equality axioms, and we deal with it later
after we introduce the definition of proper substitution - see stdpc4 1520.
The relationship of this axiom to other predicate logic axioms is
different than in the classical case. In particular, the current proof of
ax4 1892 (which derives ax-4 1316
from certain other axioms) relies on ax-3 716
and so is not valid intuitionistically. (Contributed by NM,
5-Aug-1993.) |